# Comparing NumPy Arrays with Tolerance for Optimal Results

## How to Compare NumPy Arrays for Optimal Results

NumPy, short for Numerical Python, is a popular library for Python programming to perform numerical computations, making it an essential tool for scientific computing and analysis. NumPy arrays, the primary data structure used by the library, are designed for efficient mathematical operations, but sometimes, we need to compare arrays to verify that our calculations are correct or to perform tests.

In this article, we will explore two useful methods to compare NumPy arrays, discussing their strengths, weaknesses, and applications.

### Method 1: Testing Element-wise Equality

The first method we will cover is testing element-wise equality.

This method checks if all the corresponding elements of two arrays are equal. If all values match, the method returns true; otherwise, it returns false.

### Let us consider the example below

``````import numpy as np
a = np.array([1, 2, 3])
b = np.array([1, 2, 3])
c = np.array([4, 5, 6])
print(np.array_equal(a,b)) #True
print(np.array_equal(a,c)) #False
``````

In the given example, `np.array_equal()` is used to compare the arrays `a` and `b`. Since every element of both arrays is equal, the method returns `True`.

On the other hand, when we compare the arrays `a` and `c` using the same method, we get `False` as both arrays have different elements.

### Strengths:

The primary benefit of this method is that it checks for absolute equality between the two arrays in a straightforward manner. It is useful for applications where precise matches are necessary, such as testing mathematical functions and algorithms.

### Weaknesses:

One significant limitation of this method is that it does not account for the decimal point difference. For instance, let us compare the following arrays using `np.array_equal()` method.

``````import numpy as np
x = np.array([1.1, 2.2])
y = np.array([1.11, 2.21])
print(np.array_equal(x,y)) #False
``````

In this example, the `np.array_equal()` method will return `False`, even though the arrays are almost equal. Hence it is not ideal for comparing floating-point arrays.

### Method 2: Testing Element-wise Equality Within Tolerance

The second method is testing element-wise equality within a certain tolerance level using the `allclose()` function. This method checks if all corresponding elements of two arrays are equal within a given tolerance level.

If all elements are approximately equal, the function returns `TrueFalse`.

### Let us look at the example below:

``````import numpy as np
a = np.array([1.0, 2.0, 3.0])
b = np.array([1.01, 2.02, 3.0])
print(np.allclose(a,b, rtol=0.02, atol=0.02)) #True
``````

In the example, `np.allclose()` method is used to find if two arrays are equal within a given tolerance level of 0.02 using relative tolerance and absolute tolerance respectively. Here, since both arrays have the same elements, within a certain tolerance level, the method returns `True`.

### Strengths:

This method is more useful than `np.array_equal()` method for comparing floating-point arrays, where we can tolerate slight errors while still considering the arrays equal. By specifying the absolute and relative tolerances, we can adjust the sensitivity of the method to meet our requirements.

### Weaknesses:

However, this method can produce incorrect output for arrays with different sizes or shapes. It is necessary to use this method only for comparing arrays that match in size and shape.

## Conclusion

In conclusion, both `np.array_equal()` and `np.allclose()` methods are useful for comparing NumPy arrays in different situations. While `np.array_equal()` is ideal for comparing arrays where precise matches are necessary, `np.allclose()` can be used for testing almost equal arrays with tolerances for floating-point arrays.

By using these methods, we can track if our calculations are accurate and reliable.

Expanding on our previous discussion of comparing NumPy arrays, we will now delve deeper into the second method: testing element-wise equality within a certain tolerance level using the `allclose()` method.

While comparing arrays for element-wise equality is straightforward, comparing arrays containing floating-point numbers requires additional consideration. Floating-point numbers are stored in a binary format, making it challenging to represent every value accurately.

Due to their representation in the computer, slight errors can occur when performing mathematical operations, leading to small differences in the results. These differences in values can make direct comparisons between two arrays for equality challenging.

`np.allclose()` method provides a solution to this problem by comparing arrays within a certain tolerance zone concerning the given tolerance level. To make it a little more clear, let us understand the tolerances in more detail.

### What are Tolerance Levels?

A tolerance zone is an acceptable range of values that are allowed to differ between two arrays without considering them unequal.

For example, let us consider two floating arrays `a` and `b` to the same length containing floating-point numbers.

``````import numpy as np
a = np.array([1.264, 2.047, 3.338])
b = np.array([1.263, 2.042, 3.337])
``````

Here, `a` and `b` are different but similar arrays. Directly comparing them may result in a `False` value when checking for element-wise equality.

``````print(np.array_equal(a,b)) #False
``````

However, if we now calculate the differences between every corresponding pair of elements of the two arrays:

``````delta = a - b

print(delta)
# array([0.001, 0.005, 0.001])
``````

Here, we can see that nearly all elements fall close to each other and differ only within an acceptable range of tolerance. Hence, to test the element-wise equality of these two arrays while still accounting for possible errors due to floating-point arithmetic inconsistencies, we define both an absolute and relative tolerance difference between the arrays.

### The following formula shows how to mathematically calculate the tolerance differences between two arrays:

``````tol = abs(b-a)
atol = min([min(a), min(b)])*0.01
``````

In this calculation, `tol` computes the relative difference tolerance difference, with `0.01` being the default relative tolerance value. Similarly, `atol` computes the absolute tolerance difference.

`rtol` (or `rel`) and `atol` (or `abs`) are optional inputs in the `np.allclose()` function. By default, `rtol` is equal to a special relative tolerance constant that depends on the type of the floating-point number (single, double, or long double) in the array.

This default value is `1.0e-08`, or 1 part in 10^8. Similarly, `atol` is equal to `1.0e-05`, or 1 part in 10^5 of the smallest value in the array.

For large and complex arrays, these default tolerance values may not be sufficient. Therefore, it is good practice to specify the parameters `rtol` and `atol` explicitly for better control over tolerances.

### Let us look at an example:

``````import numpy as np
a = np.array([1.2, 2.0, 3.989])
b = np.array([1.3, 2.1, 4.0])
print(np.allclose(a, b, rtol=0.02, atol=0.01)) # True
``````

In this example, when the tolerance levels are explicitly stated as `0.02` for `rtol` and `0.01` for `atol`, we get `True`, meaning that the two arrays are equal within the given tolerance levels.

### Strengths:

The primary advantages of using `np.allclose()` for comparing floating-point arrays is that it is flexible, allows for customization of the tolerance range, and helps prevent inaccuracies caused by inconsistent floating-point arithmetic.

It is also efficient, and the calculation of tolerance is done with vectorized operations, meaning that it can handle both small and large arrays quickly.

### Weaknesses:

The main disadvantage of using `np.allclose()` is that it can lead to incorrect conclusions when comparing arrays that are nearly equal.

It also fails when comparing arrays with different sizes and shapes.

## Conclusion

In conclusion, the use of tolerance in comparing NumPy arrays can be highly beneficial when comparing floating-point arrays. Choosing the right tolerance levels is essential, as incorrect tolerance values can lead to false positives and false negatives.

However, carefully adjusting the tolerance parameters can lead to substantial improvements in the accuracy of the calculations. When in doubt, it is always a good idea to test your code with different tolerance levels to verify that the results are still valid.

In conclusion, comparing NumPy arrays in Python is an essential skill for data scientists, computer programmers, and other professionals performing numerical computations. While the `np.array_equal()` method checks for absolute equality without considering floating-point errors, the `np.allclose()` method is a more sophisticated way of comparing arrays within the tolerance range, which is helpful for floating-point arrays.

Choosing the right tolerance levels is essential, as incorrect tolerance values can lead to false positives and false negatives. It is always a good idea to test your code with different tolerance levels to verify that the results are still valid.

By using these methods, we can ensure the accuracy and reliability of our calculations, increasing their quality and usefulness.